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The Complete Standard Model Lagrangian & Fermion Spectrum

by markwh04@yahoo.com
Tags: fermion, lagrangian, model, spectrum, standard
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markwh04@yahoo.com
#1
Apr4-07, 05:00 AM
P: n/a
The Standard Model Lagrangian
http://federation.g3z.com/Physics/in...#StandardModel

The Lagrangian for the Standard Model is written out in full, here.

I've made some additions from the previous version (e.g. the
representation of the gauge generators in the "Casimir Basis") and a
few corrections.

The primary novelty of the approach adopted here is the deeper
analysis of the fermionic space.

(Graphic depiction of the "Isocolor Lattice" and "Fermion Cube"
included)

Analogous to the situation in the 19th century in which Maxwell
inserted the "displacement current" term in the field law for
electromagnetism in order to retain a charge conservation law and
bring out the symmetric structure of the equations, the right
neutrinos play the corresponding role in the present situation. Here,
the symmetric structure that emerges is that, with the inclusion of
the extra terms, the fermion space factors significantly. By employing
this symmetric structure, the Lagrangian may be written in a
substantially more transparent fashion. Two bases for fermion space
will be developed here: the hypercolor basis and the Casimir basis.
The Standard Model, itself, is included as a special case within an
enveloping generalization of Yang-Mills-Higgs theories that provides
room for future extension.

The Yukawa sector is developed from first principles and the
conversion from the charge to mass eigenstates is worked out in
detail. Some notes have also been added regarding how the mass terms
for the right neutrino sector would be incorporated.

Finally, the terms involving gravitational interaction are included. I
haven't yet included any material on the issue of anomalies. An
additional anomaly cancellation arises that requires the trace of all
the gauge generators to be zero.

For the minimal standard model (i.e. without right neutrinos) there is
an effective Baryon number violation. These are issues I'm not (yet)
too familiar with.

Also an indirect mention of Tsou's "dualized Standard Model" theory
was made in the discussion of section 3.5. At the time I wrote this, I
didn't know there was such a thing, so I haven't yet expanded on this
too much. But further discussion is slated for inclusion. The dualized
Standard Model, for those who don't know, is the closest thing there
is to a theoretical explanation of the mass matrix and generational
structure. It effectively treats the 3-fold generational degeneracy as
an dual to the color SU(3) 3-fold color degeneracy. (I raise the
question in the discussion about whether the 3+1 quark+lepton
structure is more fully dualized, as well -- i.e. a sterile 4th
generation).

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markwh04@yahoo.com
#2
Apr7-07, 05:00 AM
P: n/a
On Apr 3, 5:08 pm, markw...@yahoo.com wrote:
> The Standard Model Lagrangianhttp://federation.g3z.com/Physics/index.htm#StandardModel
>
> The Lagrangian for the Standard Model is written out in full, here.
>
> The primary novelty of the approach adopted here is the deeper
> analysis of the fermionic space.
>
> (Graphic depiction of the "Isocolor Lattice" and "Fermion Cube"
> included)


More information on the Boson spectrum is also included, along with a
graphical depiction of their spectra (the "Iso-Boson Lattice" and
"Gluon Lattice" diagrams).

> The Yukawa sector is developed from first principles and the
> conversion from the charge to mass eigenstates is worked out in
> detail. Some notes have also been added regarding how the mass terms
> for the right neutrino sector would be incorporated.


The way this works is that *assuming* a particular spectrum for the
scalar fields (here, (2,1)_{Y=1/2}), and assuming it couples
trilinearly with the fermion in such a way that the trilinear term is
Hermitean and gauge invariant, one can then *derive* the particular
form it must take -- that is, one can prove it must be a Yukawa term.

For more involved scalar representations, this will not readily
generalize. An essential use is made of the fact that the Higgs sector
is irreducible. Without irreducibility, things get much more
complicated.

> Also an indirect mention of Tsou's "dualized Standard Model" theory ...
> The dualized Standard Model ... is the closest thing there is to a
> theoretical explanation of the mass matrix and generational
> structure. It effectively treats the 3-fold generational degeneracy as
> an dual to the color SU(3) 3-fold color degeneracy.


This is worth describing in a little more detail here. The model is
not readily accessible to mainstream Physicists because it makes use
of the Yang-Mills generalization of the electric/magnetic duality of
abelian fields. The problem with generalizing duality to non-abelian
fields is that the potential explicitly appears in the field
equations, so the equations themselves are not dual-invariant. It
turns out, though, that one can still frame a definition for electric/
magnetic duality -- but only within the language of the loop
representation of gauge theory. As of yet, there is no known local
spacetime formulation for this duality.

t'Hooft showed that the dual fields of a Yang-Mills theory must be
such that one is confined if and only if the other is an unbroken
symmetry. This is actually the basis of most attempts to explain quark
confinement and has been for the past 30 years since t'Hooft's
theorem. What Tsou does is identify the dual of SU(3) with
"generational" SU(3).

The exact symmetry of color SU(3) has, as its counterpart, the broken
symmetry of generational SU(3). The corresponding scalar "Higgs"
fields are 3 pairs of color triplets whose vacuum values are (up to
scaling) (x,0,0), (0,y,0) and (0,0,z) where (x,y,z) lies on the unit
sphere. The rank-1 matrix formed of (x,y,z)(x,y,z)^T is identified as
the mass matrix!

This carries with it, the assertion that the mass matrix for each
sector (charged lepton, uncharged lepton, up-type quarks, down-type
quarks) are each of rank 1. This rank 1 factoring survives the running
of the couplings corresponding to the extra Higgs structure. Thus, the
mass matrices of the standard model (by which I mean the CKM and MNS
matrix plus the diagonal matrices comprising the mass values
themselves) are derived by an appropriate running of the couplings.
The fit to experimental values is VERY close. One parameter, as of the
2003 paper, is off; but since all the calculations are preliminary
there's room for further refinement of the model to accommodate a
better fit. Ultimately, everything comes out of only 3 parameters --
the two comprising the unit sphere which (x,y,z) lie on, and a third
representing the scaling of the extra Higgs fields.



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