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

1W1+ 2W2 + 3W3

Substitute B = cosW A + sin W Z0

Sum over first generation particles.

Flavor changing interactions.

Left handed only

Flavor up Flavor down

up down

Weak interaction terms

flavor changing: leptons flavor 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

Z0

Z0

Consider only the A term:

ea1 ea2

gives agreement with experiment.

Cf = 2T3

= -1

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

A

Z0

(E & M) QED interactions

weak neutral current interactions

weak flavor changing interactions

QCD color interactions

-

+

The Standard Model Interaction Lagrangian for the 1st generation

Weak neutral current interactions

Z0

Z0

Z0

Z0

quarks

leptons

Weak charged flavor changing interactions

g2

g2

Quantum Chromodynamics (QCD): color forces

Only non-zerocomponents 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 ggrg-ggb

-

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

ud

u

d

d

W doesn’t see color

u

W-

decay of -

-

u

d

-

p

p

duu

uud---

W production from

p-

p

p p

-

-

W+

The nuclear force

np

u

d

d

u

d

u

u

u

d

u

pn

d

d

u

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 function right vertex function

Mfi =

spinspin

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 aa function of the momenta only, and the four spin (helicity) states.

coupling constant –one for each vertex

Note that each vertex isgenerated by the interactionLagrangian density.

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. Theseterms lead to the interaction of gluons with other gluons.

G G

quarkloop

gluonloop

NfNc

Nf= # flavors Nc= # colors

normal free gluon term3-gluon vertex

Note sign

momentum squared of exchanged gluon

Nf Nc

Nc

Nf

In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign.

-7

M2quark

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 generatesterms in the Lagrangian density which produce 3-gluon vertices – and gluonloops in the exchanged gluon “propagator”.

The Higgs Lagrangian Contribution