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The Complete Standard Model Lagrangian & Fermion Spectrum 
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#1
Apr407, 05:00 AM

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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 YangMillsHiggs 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 3fold generational degeneracy as an dual to the color SU(3) 3fold 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). 


#2
Apr707, 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 "IsoBoson 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 3fold generational degeneracy as > an dual to the color SU(3) 3fold 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 YangMills generalization of the electric/magnetic duality of abelian fields. The problem with generalizing duality to nonabelian fields is that the potential explicitly appears in the field equations, so the equations themselves are not dualinvariant. 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 YangMills 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 rank1 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, uptype quarks, downtype 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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