Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons  1 W 1 +  2 W 2 +  3 W 3

Substitute B  = cos  W A  +  sin  W Z 0  Sum over first generation particles. Flavor changing interactions. Left handed only Flavor up Flavor down updown

Weak interaction terms flavor changing: leptonsflavor changing: quarks

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

Consider only the A  term: ea1ea1 ea2ea2 gives agreement with experiment. C f = 2T 3  = -1

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

AA Z0Z0

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

Weak neutral current interactions Z0Z0 Z0Z0 Z0Z0 Z0Z0

quarks leptons Weak charged flavor changing interactions g2g2 g2g2

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

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

The QCD interaction Lagrangian density

The red, anti-green gluon The green, anti-blue gluon Note that there are only 8 possibilities: r gr g grggrg - ggbggb -

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

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

decay of  - -- u d -

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

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

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.

Feynman rules applied to a 2-vertex electron positron scattering diagram left vertex functionright vertex function M fi = spin time propagator metric tensor 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. coupling constant – one for each vertex Note that each vertex is generated by the interaction Lagrangian density.

Confinement of quarks free quark termsfree gluon termsquark- 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.

G  quark loop gluon loop NfNf NcNc N f = # flavorsN c = # colors normal free gluon term 3-gluon vertex Note sign

momentum squared of exchanged gluon NfNf NcNc NcNc NfNf In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign. -7  M 2 quark

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”.

The Higgs Lagrangian Contribution