Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons

1. Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons  1W1+  2W2 +  3W3

2. Substitute B = cosW A  + sin W Z0  Sum over first generation particles. up down Left handed only Flavor up Flavor down Flavor changing interactions.

3. Weak interaction terms flavor changing: leptons flavor changing: quarks

4. We want the coefficient for the electron-photon term to be -e f=0 for neutrino and = 1 for others A -e A Z0 Z0

5. Consider only the A term: ea1 ea2 gives agreement with experiment. Cf = 2T3  = -1

6. The following values for the constants gives the correct charge for all the particles.

7. A Z0

8. The Standard Model Interaction Lagrangian for the 1st generation (E & M) QED interactions weak neutral current interactions - weak flavor changing interactions + QCD color interactions

9. Weak neutral current interactions Z0 Z0 Z0 Z0

10. Weak charged flavor changing interactions g2 quarks g2 leptons

11. Quantum Chromodynamics (QCD): color forces Only non-zero components of  contribute.

12. To find the final form of the QCD terms, we rewrite the above sum, collecting similar quark “color” combinations.

13. The QCD interaction Lagrangian density

14. Note that there are only 8 possibilities: grg - rg The red, anti-green gluon - ggb The green, anti-blue gluon

15. The gluon forces hold the proton together At any time the proton is color neutral. That is, it contains one red, one blue and one greenquark. proton

16. beta decay u u d d d u W- proton neutron W doesn’t see color

17. decay of - - u d -

18. - Wproduction from - p p p d u p u W+ - d - u - p p - u

19. The nuclear force u u d d n p d u u W- u d p d d n u u - Note that W-  d + u = -In older theories, one would consider rather the exchange of a - between the n and p.

20. Cross sections and Feynman diagrams everything happens here transition probability amplitude must sum over all possible Feynman diagram amplitudes with the same initial and final states .

22. Feynman rules applied to a 2-vertex electron positron scattering diagram Note that each vertex is generated by the interaction Lagrangian density. time spin spin metric tensor Mfi = left vertex function right vertex function coupling constant – one for each vertex propagator The next steps are to do the sum over  and  and carry out the matrix multiplications. Note that is a 4x4 matrix and the spinors are 4-component vectors. The result is a a function of the momenta only, and the four spin (helicity) states.

23. Confinement of quarks free quark terms free gluon terms quark- gluon interactions The free gluon terms have products of 2, 3 and 4 gluon field operators. These terms lead to the interaction of gluons with other gluons.

24. G G Note sign normal free gluon term 3-gluon vertex Nf= # flavors Nc= # colors Nc Nf quark loop gluon loop

25. momentum squared of exchanged gluon Nf Nc  M2quark Nf Nc -7 In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign.

26. Quark confinement arises from the increasing strength of the interaction at long range. At short range the gluon force is weak; at long range it is strong. This confinement arises from the SU(3) symmetry – with it’s non-commuting (non-abelian) group elements. This non-commuting property generates terms in the Lagrangian density which produce 3-gluon vertices – and gluon loops in the exchanged gluon “propagator”.

27. The Higgs Lagrangian Contribution